Complemented subspaces of a $C(K)$-space constructed by Candido

TL;DR

Under Ostaszewski's club-suit principle, Candido's locally compact space $L$ yields a $C_0(L \times L)$ with a canonical operator decomposition into parts $pI$, $qJ$, $M_g$, $N_h$, a finite-range product $R_1R_3R_2R_4$, and $S$. The authors classify complemented subspaces by a symmetric/antisymmetric decomposition and a case analysis on the four possible forms of the $Z_1^\rho$-action, showing that any complemented subspace must be isomorphic to one of a finite list, including $0$, $\mathbb{R}^n$, $c_0$, $C_0(\omega^\omega)$, $C_0(L)^n$, $C_0(\omega \times L)$, $C_0(\omega^\omega \times L)$, $C_0(\Omega_1)$, $C_0(\Omega_2)$, or $C_0(L \times L)$. Consequently, the complemented subspaces of $C_0(\Omega_j)$ are precisely listed as $0$, $\mathbb{R}^n$, $c_0$, $C_0(\omega^\omega)$, $C_0(L)^n$, $C_0(\omega \times L)$, $C_0(\omega^\omega \times L)$, and $C(\Omega_j)$ for $j=1,2$. This extends the understanding of operator structures and subspace classifications in exotic $C(K)$-spaces under set-theoretic assumptions and yields a complete descriptive classification in this setting.

Abstract

We classify the complemented subspaces of $C_0(L \times L)$, where $L$ is an exotic locally compact Hausdorff space recently constructed by Candido under Ostaszewski's $\clubsuit$-principle.

Figures (1)

• Figure 1: Partition of $L \times L$ in disjoint clopen sets.

Theorems & Definitions (22)

• Theorem 1.1: $\clubsuit$
• Theorem 1.2: Candido candido2024few
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3: Candido
• proof
• Definition 2.4
• Remark 2.5
• Lemma 2.6